Some Fundamentals of Stability
Theory
Aaron Greenfield
Outline
I.
II.
III.
IV.
Introduction + Motivation
Definitions
Theorems
Techniques for Lyapunov Function
Construction
Basic Notion of Stability
Stability
An important property of dynamic systems
Stability. . .
An “insensitivity” to small perturbations
Perturbations are modeling errors of system, environment, noise
F=0. OK
Basic Notions of Stability
Stability
An important property of dynamic systems
Stability. . .
An “insensitivity” to small perturbations
Perturbations are modeling errors of system, environment
F=0. OK
Basic Notions of Stability
Stability
An important property of dynamic systems
Stability. . .
An “insensitivity” to small perturbations
Perturbations are modeling errors of system, environment
F=0. OK
Basic Notions of Stability
Stability
An important property of dynamic systems
Stability. . .
An “insensitivity” to small perturbations
Perturbations are modeling errors of system, environment, unmodeled
noise
F=0. OK
Basic Notions of Stability
Stability
Why might someone in robotics study stability?
(1) To ensure acceptable performance of the robot under perturbation
  Mq  Cq  g
Configuration space trajectory
with constraints
Some Notation
xe
p(t ; x0 , t0 )
|| x ||
An isolated equilibrium of an ODE
A solution curve to first-order ODE system
with initial conditions listed
Standard Euclidean Vector Norm
Definitions
MANY definitions for related stability concepts
Restrict attention to following classes of differential equations
x  f (x)
Autonomous ODE
x  f ( x, t )
Non-Autonomous ODE
x  f ( x, u, t )
Reduces to above under action of a control
Stabilizability Question
Definitions Summary Slide
Attractivity
Lyapunov Stability
x  f (x)
With Isolated equilibrium at
xe  0
Defn1.1: Stability of autonomous ODE, isolated equilibrium
The equilibrium point (or motion) is called stable in the sense
of Lyapunov if:
For all   0 there exists
 0
such that
|| p(t; x0 , t0 ) ||   t  t0
whenever ||
x0 ||   ( )
p(t; x0 , t0 )
Hahn 1967
Slotine, Li
Lyapunov Stability
x  f (x)
With Isolated equilibrium at
xe  0
Defn1.1: Stability of autonomous ODE, isolated equilibrium
The equilibrium point (or motion) is called stable in the sense
of Lyapunov if:
For all   0 there exists
 0
|| p(t; x0 , t0 ) ||   t  t0
whenever ||
x0 ||   ( )
such that
Notes
1    1 (Local Concept)
(2) There can be a  max but no  min
(1) If
(Unbounded Solutions)
Lagrange Stability
x  f (x)
With Isolated equilibrium at
xe  0
Defn1.1: Stability of autonomous ODE, isolated equilibrium
The equilibrium point (or motion) is called stable in the sense
of Lyapunov if:
For all   0 there exists
 0
|| p(t; x0 , t0 ) ||   ( ) t  t0
whenever
|| x0 ||  
such that
Lagrange Stability
x  f (x)
With Isolated equilibrium at
xe  0
Defn1.1: Stability of autonomous ODE, isolated equilibrium
The equilibrium point (or motion) is called stable in the sense
of Lyapunov if:
For all   0 there exists
 0
such that
|| p(t; x0 , t0 ) ||   ( ) t  t0
whenever
|| x0 ||  
Lagrange Stable
Lagrange Stability
x  f (x)
With Isolated equilibrium at
xe  0
Defn1.1: Stability of autonomous ODE, isolated equilibrium
The system is Lagrange stable if:
For all 
 0there exists   0
|| p(t; x0 , t0 ) ||   ( ) t  t0
whenever || x0 ||  
such that
Notes
(1) Bounded Solutions
(2) Independent Concept
a) Lyapunov, Lagrange
b) Not Lyapunov, Lagrange
c) Lyapunov, Not Lagrange
d) Not Lyapunov, Not Lagrange
Attractive
x  f (x) With Isolated equilibrium at xe  0
Defn 1.2: Attractivity of autonomous ODE,isolated equilibrium
The equilibrium point (or motion) is called attractive if:
There exists an   0
lim p (t ; x0 , t0 )  xe
t 
such that
whenever
x 0 :|| x0 ||  
Notes
(1) Asymptotic concept, no transient notion
(2) Stability completely separate concept
a) Stable, Attractive
b) Unstable, Unattractive
c) Stable, Unattractive
d) Unstable, Attractive
(3) Unstable yet attractive, Vinograd
Attractivity Example
x  f (x) With Isolated equilibrium at xe  0
Defn 1.2: Attractivity of autonomous ODE,isolated equilibrium
x 2 ( y  x)  y 5
x  2
( x  y 2 )(1  ( x 2  y 2 ) 2 )

x  0 if x  0; y  0
y 2 ( y  2 x)
y  2
( x  y 2 )(1  ( x 2  y 2 ) 2 )

y  0 if x  0; y  0
Denominator always positive
y Switches on
y  2x
Asymptotic Stability
x  f (x)
With Isolated equilibrium at
xe  0
Defn 1.3: Asymptotic stability of autonomous ODE,
isolated equilibrium
Asymptotically stable equals both stable and attractive
Defn 1.4: Global Asymptotic stability of autonomous ODE,
isolated equilibrium
Global Asymptotic Stability is both stable and attractive for 
[Hahn]
Set Stability
Now consider stability of objects other than isolated equilibrium point
Defn 1.5: Stability of an invariant set M,
autonomous ODE
The set M is called stable in the sense of Lyapunov if:
For all   0 there exists
 0
such that
ρ(p(t), M)  ε t  t0
whenever ρ(x 0 , M)  δ
Invariant-Not entered or exited
Notes
(1) Attractivity, Asymptotic Stability
are comparably redefined
(2) Use on limit cycles, for example
 ( x, M )  inf || x  y ||, y  M
[Hahn]
Motion Stability
Now consider stability of objects other than isolated equilibrium point
Defn 1.6: Stability of a motion (trajectory),
autonomous ODE
The motion
p(t; x0 , t0 ) is stable if:
For all   0 there exists
 0
such that
|| p(t; x0 , t0 )  p(t; x1 , t0 ) ||   t  t0
whenever
|| x1 - x0 ||  
Notes
(1) Just redefined distance again
(2) Error Coordinate Transform
[Hahn]
Uniform Stability
x  f ( x, t )
With Isolated equilibrium at
xe  0
Defn2.1: Stability of non-autonomous ODE,
isolated equilibrium
The equilibrium point (or motion) is called stable (Lyapunov) if:
For all   0 there exists
 0
such that
|| p(t; x0 , t0 ) ||   t  t0
whenever
|| x0 ||   ( , t0 )
Defn 2.2: Uniform Stability of non-autonomous ODE,
isolated equilibrium
|| x0 ||   ( , t0 )
 || x0 ||   ( )
Definitions
x  f ( x, t )
With Isolated equilibirum at
xe  0
Defn2.1: Stability of non-autonomous ODE,
isolated equilibrium
Stable, not uniformly stable system
x  (t sin t  cos t  2) x
x(t )  x0 exp t0 (2  cos t0 )* exp  t (2  cos t )
[Dunbar]
Definitions-Wrap Up Slide
Autonomous ODE
Stability of Equilibrium
Lagrange Stability
Attractivity
Asymptotic Stability
Stability of Set
Stability of Motion
Not Covered:
Non-Autonomous ODE
Same
Uniform Stability
Exponential Stability
Input-Output Stability BIBO-BIBS
Stochastic Stability Notions
Stabilizability, Instability, Total
Theorems
How do we show a specific system has a stability property?
MANY theorems exist which can be used to prove some stability property
Restrict attention again to autonomous, non-autonomous ODE
These theorems typically relate existence of a particular function
(Lyapunov) function to a particular stability property
Theorem:
If
then
there exists a Lyapunov function,
some stability property
Lyapunov Functions
Lyapunov Functions
Defn 3.1 Lyapunov function for an autonomous system
V (x )
V
V ( x) 
 f 0
x
Positive Definite around origin
x  f (x)
V ( x)  0  x  0
V ( x)  0
For some neighborhood of origin
Defn 3.2 Lyapunov function for an non-autonomous system
V ( x, t )  V2 ( x)t
V
V
V ( x, t ) 
f
0
x
t
x  f ( x, t )
Dominates Positive Definite Fn
For some neighborhood of origin
Note
Assume
V is continuous in x,t
V is also
[Slotine, Li]
[Hahn]
Stability Theorem
Thm 1.1: Stability of Isolated Equilibrium of Autonomous ODE
An isolated equilibrium of x 
for this system
is stable if there exists a Lyapunov Function
Proof Sketch 1.1
If
(1) Pick Arbitrary Epsilon, Construct Delta
V ( x), V ( x)  0 exists
(2) Consider min of V(x) on || x ||  Vbound
Extreme Value Theorem
then
For all   0 there exists 
0
|| p(t; x0 , t0 ) ||   t  t0
whenever
f (x)
x  f (x)
|| x0 ||  
(3) Define function
f ( )  max V ( x) :|| x || 
(4) If
f ( )
continuous, then by IVT
  (0,  ) : f ( )  vbound
(5) Since V ( x)  0
|| x0 ||  || x(t ) || 
Stability proof example
Thm 1.1: Stability of Isolated Equilibrium of Autonomous ODE
An isolated equilibrium of x 
for this system
f (x)
is stable if there exists a Lyapunov Function
Example- Undamped pendulum
     


  sin   0       sin  

  
(1) Propose
1
V ( ,  )  1  cos     2
2
(Kinetic + Potential)
(2) Derivative
x  f (x)
V ( ,  )  sin  *    0
Asymptotic stability theorem
Thm 1.2: Asymptotic Stability of Isolated Equilibrium of Autonomous ODE
An isolated equilibrium of x  f (x) is asymptotically stable if there exists
a Lyapunov Function for this system with strictly negative time derivative.
Small Proof Sketch 1.2
(1) Stability from prev, Need Attractivity
V ( x)
(2) EVT with
 :|| p(t; x0 ) ||  t

V ( x)
“Ball” not entered
(3) Construct a sequence of Epsilon balls
Notes
Local
V ( x)  0 locally
Global
V ( x)  0 globally
V ( x)  , || x || 
Radial Unbounded, Barbashin Extension
Lasalle Theorem
Thm 1.3: Stability of Invariant Set of Autonomous ODE
(Lasalle’s Theorem)
x  f (x)
Use: V ( x)  0 and Limit Cycle Stability
Let there be a region Gl be defined by: x : V ( x)  l
Let V ( x)  0 on G
l
E : V 1 (0);
M is largest invariant
Let there be two more regions E and M:
M  E  Gl set
 ( p (t ), M )  0
Then M is attractive, that is lim
t 
Small Proof Sketch 1.3
(1) Define Positive Limit Set
( x0 )  { p : {tn } and p(tn ; xo )  p as tn  }
Properties:
Invariant, Non-Empty, ATTRACTIVE!!
(2) Show ( x0 )  V 1 (0)  E
[Lasalle 1975]
Lasalle Theorem example
Lasalle’s Theorem Example
Example- Damped pendulum



     sin   b 
   



  b  sin   0     
(1) Propose
(2) Derivative
1
V ( ,  )  1  cos     2
2
V ( ,  )  sin  *  
 b 2  0
E  V 1 (0)  {( ,   0)}

M : (  0,   0)
Asymptotic Stability of Origin
Uniform Stability Theorem
Theorems for Non-Autonomous ODE
x  f ( x, t )
Stability and Asymptotic Stability remain the same
V ( x, t )  0  Stability
V ( x, t )  0  Asymptotic Stability
Thm 1.4: Uniform (Stability) Asymptotic Stability of Non-Autonomous ODE,
Isolated Equilibrium point
The equilibrium is uniformly (Stable) asymptotically stable if there exists
A Lyapunov function with (V ( x, t )  0) V ( x, t )  0 and there exists
a function such that:
V ( x, t )  V1 ( x)
Decrescent
Small Proof Sketch 1.4
 (|| x ||)  V ( x, t )   (|| x ||)
Positive Definite and Decrescent
 ( )   ( )  V ( x, t )   (|| x(t ) ||)
 ( )   (|| x(t ) ||)
[Slotine,Li]
Barbalet’s Lemma
Thm 1.5: Barbalet’s Lemma as used in Stability
(Used for Non-Autonomous ODE) x

f ( x, t )
If there exists a scalar function V ( x, t ) such that:
(1) V has a lower bound
(2) V  0
(3) V ( x, t ) is uniformly continuous in time
Then lim V ( x, t )  0
t 
V ( ,  )  sin  *  
 b 2  0
Barbalet
 lim  (t )  0
t 
[Slotine,Li]
Theorems-Wrap Up Slide
Autonomous ODE
Non-Autonomous ODE
Lyapunov implies stability Same
Lyapunov implies a.s
Uniform Stability
Lasalle’s Theorem for sets Barbalet’s Lemma
Not Covered:
Instability Theorems
Converse Theorems
Stabilizability
Kalman-Yacobovich, other Frequency
theorems
Techniques for Lyapunov Construction
Theorems relate function existence with stability
How then to show a Lyapunov function exists? Construct it
In general, Lyapunov function construction is an art.
Special Cases
Linear Time Invariant Systems
Mechanical Systems
Construction for Linear System
Construction for a Linear System
x  Ax
V ( x)  xT Px
(1) Propose
(2) Time Derivative
P is symmetric
P is positive definite
V ( x)  V  Ax  2 xT PAx
 xT ( PA  AT P) x
If we choose
Q0
and solve algebraically for P:
PA  AT P  Q
As long as A is stable, a solution is known to exist.
Also an explicit representation of the solution exists:

P   e Qe At dt
AT t
0
Construction for a Mechanical System
Construction for a Mechanical System
V (q, q ) 
  M (q)q  Cq  g (q)
1 T
1
q M (q)q  q T K p q
2
2
Kinetic Energy
(1) Propose
(or similar)
Potential Energy
1
V (q, q )  q T M (q )q  q T M (q )q  q T K p q
2
(2) Time Derivative
If we use PD-controller with gravity compensation
  K p q  K d q  g
then
V (q, q )  q T K D q
Asymptotically stable with Lasalle
[Sciavicco,Siciliano]
General Construction Techniques
Construction methods for an Arbitrary System
x  f (x)
Krasofskii
V ( x)  f T Pf
A quadratic form (ellipsoid) of system velocity
f
f
V ( x)  f Pf  f Pf  f (
*P  P* ) f
x
x
T
T
Solve
T
T
f
f
*P  P*  Q  0
x
x
T
Variable Gradient
x
V ( x)   Vdx
0
Assume a form for the gradient, i.e Vi 
Solve for negative semi-definite gradient
Integrate and hope for positive definite V
n
a x
j 1
ij
j
[Slotine, Li]
[Hahn]
Construction Wrap-Up Slide
(1) Linear System -> Explicity Solve Lyapunov Equation
(2) Mechanical System -> Try a variant of mechanical energy
(3) Krasovskii’s Method
Variable Gradient
Problem specific trial and error
Conclusion




Motivated why stability is an important
concept
Looked at a variety of definitions of various
forms of stability
Looked at theorems relating Lyapunov
functions to these notions of stability
Looked at some methods to construct
Lyapunov functions for particular problems